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Bounds on the iterations of a function!

Source: Romania TST 2014 Day 2 Problem 2

January 21, 2015
functionalgebra unsolvedalgebra

Problem Statement

Let aa be a real number in the open interval (0,1)(0,1). Let n≥2n\geq 2 be a positive integer and let fn ⁣:R→Rf_n\colon\mathbb{R}\to\mathbb{R} be defined by fn(x)=x+x2nf_n(x) = x+\frac{x^2}{n}. Show that a(1−a)n2+2a2n+a3(1−a)2n2+a(2−a)n+a2<(fn∘ ⋯ ∘fn)(a)<an+a2(1−a)n+a\frac{a(1-a)n^2+2a^2n+a^3}{(1-a)^2n^2+a(2-a)n+a^2}<(f_n \circ\ \cdots\ \circ f_n)(a)<\frac{an+a^2}{(1-a)n+a} where there are nn functions in the composition.
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